- Email: [email protected]
Modelling and simulation of a diffusion limited glucose biosensor
B
ELSEVIER
Sensors and Actuators B 33 (1996) 203-207
CHBMICAL
Modelling and simulation of a diffusion limited glucose biosensor A. Cambiaso a, L. Delfino a, M. Grattarola a,*, G. Verreschia, D. Ashworth b, A. Maines b, P. V a d g a m a b aBioelectronics Laboratory and Bioelectronic Technologies Laboratory, do Advanced Biotechnology Centre, DIBE, Universityof Genoa, via Opera Pia I IA, 16145 Genova, Italy bSection of Clinical Biochemistry, Departmentof Medicine, Hope Hospital, Eccles Old Road, Salford, M6 8HD, UK Accepted 4 March 1996
Abstract A mathematical model for an amperometric enzyme sensor is described as a tool to design glucose biosensors, operating in a diffusion limited regimen. This kind of enzyme electrode really comprises a miniature reactor system rather than a classical form of transducer. The equations constituting the model are obtained taking into account both the diffusion process and the enzymatic reaction in the general (not pseudo-stationary) case. The model is numerically integrated in order to obtain the time behaviour of the electrode current. The Michaelis-Menten apparent constant (Km app) of the whole system is obtained from the simulation and compared with that computed for a bulk system.
Keywords: Amperometric sensors; Diffusion-reactionsystems; Sensor modelling;Glucose; Computer simulations; Biosensor
1. Introduction The development of amperometric enzyme electrodes for the measurement of glucose has been the subject of intense investigation since the construction of the first electrochemical glucose sensor by Clark and Lyons in 1962 [1]. Initially, the major driving force behind this research was the need for clinical glucose monitoring, particularly in relation to the treatment of diabetes mellitus [2]. However, applications have diversified and there is a growing research effort into the monitoring of glucose, and other analytes, in industry [3], agriculture and food [4]. The oxidoreductase enzyme glucose oxidase is commonly used in the development of glucose sensors. The enzyme catalyses the oxidation of glucose: fl-D-glucose + 02 + H - O ~ Doxidase - e l u c o n""i c
acid + H202 (1)
* Corresponding author. Tel.: +39 10 3532761; fax: +39 I0 3532133; e-mail: [email protected]
0925-4005/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved Pli S0925-4005(96)01830-8
In the 'classical' device the consumption of oxygen [5] or the formation of hydrogen peroxide [6] is monitored at an electrode. The concentration of glucose in many matrices greatly exceeds the Km of glucose oxidase (4.2 mmol 1-I) and therefore limits the range over which a linear response can be achieved. This is a particular problem with food applications, where the glucose concentration can be in excess of 500 mmol I-L By the use of external substrate diffusion limiting membranes, the linearity can be greatly enhanced [7]. Amperometric detection of H202 requires a relatively high polarising voltage (ca. +0.65 V versus Ag/AgCl) and as a consequence, sensors based on this detection modality are susceptible to interference from a range of electrooxidisable species found in clinical, industrial and food matrices, including urate and ascorbate. However, interference can be prevented by the interposition of a membrane between the enzyme layer and the electrode which is capable of rejection of interferents whilst allowing diffusion of H202 [8].
204
A. Cambiaso et al./ Sensors and Actuators B 33 (1996) 203-207
2. Sensor model
A generalised glucose enzyme electrode would therefore consist of essentially three layers: the outer diffusion limiting membrane, in contact with the bulk solution containing the analyte of interest; the immobilised enzyme layer, within which substrate is depleted and product is formed; and the inner selective membrane in contact with the electrode [9]. A model of a multiiayer enzyme electrode was already proposed in Ref. [10]. In the present work, the outer membrane performs the dual function of diffusion limitation and selectivity, whilst the inner membrane is a permeable support membrane. Fig. 1 shows the modelled monodimensional structure with the basic interfaces involved in a typical electrode construction. The indicated parameters are referred to the simplified version of the reaction (I), reported in (2).
~
k
___~k+____.~
(2)
,E+P
20 0
layer I
layer3
layer2 E,C
solution
d-
I ¢leclrode
-%
__% d+ a2
X
'
0
'
1000
'
2000 Timebl
3000
Fig. 2. Electrode current transient for different substrate concentrations: (a) 10 mmol i-l, (b) 40 mmol I-!, (c) 70 mmol !-l, (d) 100 mmol! -1.
state assumptions giving the following system of nonlinear equations: aC - - - = KI+EoS-KI+CS-(KI_ + K2+)C at OS 02S - - = D s . . . . . . . KI+EoSI+KI+CS+ Kl_C
Ox 2
OP Ot
bulk
a)
40
--
The considered phenomena are: diffusion of S (substrate = glucose) and P (product) through layer 1, layer 2 and layer 3. Reaction between S and E (enzyme) in layer 2. The behaviour of the system in the different compartments is described in the following, with reference to Fig. 1: X < 0 (bulk solution): the substrate is constant (S - So) and the concentration of product is negligible (P = 0). 0 < X < XI: S and P are free to diffuse. XI
(c)
Bo
0t
E+S~-+C
(d)
100
(3)
O2P = De - -
Ox 2 +
K2+ C
where Ds and De are the diffusion coefficients for S and P in layer 2, respectively. The following boundary conditions were set at each iteration: S = SO, P = 0 in X = 0; O$/Ox =0 (that is flux = 0), P = 0 in X = X3; Ds(OPIOX) and Dp(0P/0X) constant (that is flux constant) through the other interfaces The electrode current is evaluated at the electrode surface by means of Eq. (4): i(t) = nF. A. De -a-P ;:9X (x3,t)
(4)
where A is the electrode area, De is the diffusion coefficient for P in layer 3, and nF is the total molar charge. 3. Computer simulation
The sensor model was numerically integrated by means of a computer program developed in C language on a DEC3000 workstation (Digital, Alpha). The equations related to layers 1 and 3 were discretised using the Crank-Nicholson technique [10]. Such an implicit method gives the advantage of unconditioned numerical stability in the case of simple diffusion equations. For system (3), describing layer 2, an implicit method of integration was developed, starting from the Crank-Nicholson technique. At the starting time S and P were set = 0 in the three layers; E was set equal to Eo in layer 2. 4. Resul~
Fig. l, Qualitative sketch of the one-dimensionai modelled structure.
The simulation program was tested by utilising pa-
205
A. Cambiaso et al. I Sensors and Actuators B 33 (1996) 203-207
x0
x2
xl
x3
100 80 w
60 40
g 20 0
I
I
:
I
',
:
:
:
:
,
|
One dimensional concentration profile (A.U.)
Fig. 3. Substmte concentration profile after (from bottom to top): 900 s, 1800 s, 2700 s, 3600 s.
rameter values evaluated during experiments dealing with the measurements on fruit juice. In pnrticular the diffusion coefficients have been set as follows: layer 1 5.0 x 10-I° cm 2 s-I (S), 1.0 × 10-I! cm 2 s-I (P); layer 2 1.0 x 10-Tcm 2 s-I (S), 5.0 x 10-Tcm 2 s-I (P); layer 3 1.7 x 10-9cm 2 s (S), 8.5 x 10-gem 2 s-! (P) The kinetic constants of the enzymatic reaction in layer 2 have been set: KI+ = 1 2 × 10-2 mmol -I s-I 1, KI_ = 0.68 x 10-3 s-l, /(2+ = 5.0 × 10-2 s-I. The concentration Eo in layer 2 has been set equal to 0.125 mmol 1-l. The thickness of the layers have be set: I 0 / t m for the limiting membrane, 40/~m for the enzymatic layer and 15/tm for the inner membrane. Fig. 2 shows an example of the transient behaviour of the electrode current, computed for 4 different substrate concentrations. There was a very long transient time as expected from this kind of diffusion limiting sensors. Fig. 3 shows the substrate concentration profile through the three layers constituting the model. Four curves are drawn, computed at different simulation times for a substrate concentration S = 100 mmoi i-I. The progressive diffusion of the substrate in the sensor structure can be seen.
3
x0
x2
×1
2.5
!
M
_m
2
=
1.5 =
1 0.5 0
x3
I
I
t
. ,
|
.
,
. ,
,
.
,
. •
,
.
,
.
,
In Fig. 4 the concentration profile of the product is drawn, at different simulation times, under the same conditions as Fig. 3. The slope of these curves at the electrode surface (X3) is proportional to the electrode current. The very long transient time, experimentally obtained with this kind of sensor, suggests that for the evaluation of the substrate concentration the use of techniques alternative to the measurement of the plateau current can be beneficial. For example the maximum of the derivative of the transient curves can be employed for such a purpose (see Fig. 5). The simulation program also gives the opportunity to evaluate the apparent Michaelis-Menten constant Km app taking into account the diffusion effect. The Km for the enzyme in solution is that value of the free uncombined substrate concentration at which half of the enzyme reactive sites are occupied, and the reaction velocity is 50% of the maximum. This assumes that the substrate has free access to the enzyme. If the enzyme and bulk substrate are separated by a diffusion restricting membrane the external substrate concentration can be higher before half of the enzyme sites are occupied, and the apparent Km of the enzyme in this situation is greater than in solution. This apparent Km, denoted as Km app varies as the thickness and diffusion coefficient (Dsi) of the membrane vary, and takes on a different value in each unique situa(d) 0.1
(¢)
0.08 0.06 --= 0.04 0.02
,
,
|
One dimensional concentration profile (A.U.)
0 1000
2000
3000
Time Isl Fig. 4. Product concentration profile after (from bottom to top): 900 s, 1800 s, 2700 s, 3600 s.
Fig. 5. Time derivative of the curves of Fig. 2.
206
A. Cambiaso et aL / Sensors and Actuators B 33 (1996) 203-207 (ci ~
"----'-
Acknowledgements
(a)
Work supported by the European Union, Measurements and testing Programme, BIOSUMO project.
80
References
°
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Substrate bulk Ceneentrafion |mMI Fig. 6. Steady state current curves utilised to evaluate the apparent Km, computed for three different diffusion coefficients for S on layer 1: (a) 1.1 x 10-9 cm 2 s"!, (b) g.0 x 10-I0 cm 2 s"l, (c) 5.0 x 10-10 cm2 s -!.
tion. Some simulations have been carried out in order to asses the behaviour of Km app against the variation of the bulk concentration (So). Fig. 6 shows the steady state current as a function of So, computed after 3600 s, for three different values of the diffusion coefficient Dsj of the substrate S in layer I (diffusion limiting membrane). Decreasing the diffusion coefficient can be seen to decrease the response (current) magnitude whilst extending the substrate concentration range over which there is an approximately linear relationship between response and concentration. The Km ~ parameters have been evaluated as the So value corresponding to 50% of the current maximum steady state value. Table 1 shows the calculated values for Km app as a function of layer 1 diffusion coefficient. This demonstrates that decreasing the diffusion coefficient of the outer membrane increases K m app thus extending response linearity,
$. Conclusions The proposed model is a useful tool for the design of diffusion limited amperometric sensors. The simulation program can be used to predict the effect of variation in sensor parameters on sensor response by utilising the information present in the transient period. For example, the effect of membrane thickness and permeability on response sensitivity, velocity and linearity can be predicted. Moreover, new indications can be gained about the values assumed by the kinetic constants when an enzyme reaction is forced to take place near an electrode surface,
[1] L.C. Clark and L. Lyons, Electrode system for continuous monitoring in cardiovascular surgery. Ann. iV. E Acad. ScL, 102 (1962) 29--45. [2l G.D. Velho, G. Reach and D.R. Thevenot, The design and development of in vivo glucosf~ sensors for an artificial endocrine pancreas, in I. Karube and G.S. Wilson (Eds.), Biosensors: Fundaenentals and Applicatio'zs Turner APF, Oxford University Press, New York, 1987, pp. 3cJ0--408. 13] B. Grundig and C. Krabisch, Electron mediator-modified electrode for the deterinination of glucose in fermentation media. Anal, Chim. Act~ 183 (1989) 59--66. [4] J.R. Whitaker, The need for biosensors in the food industry and food research, in G, Wagner and G.G. Guilbault (eds.), Food Bio. sensor Analysis, Marcel Dekkcr, New York, 1994, pp. 13-30. [5] SJ. Updike and G.P. Hicks, The enzyme electrode, Nature, 214 (1967) 986-988. 16] G.G. Guilbanlt, Analytical Uses of Immobilised Enzymes, Marcel Dekker, New York, 1984. [7l W.H. Mullen, F.H. Keedy, S.J. Churchouse and P.M. Vadgama, Glucose enzyme electrode with extended iinearity - application to undiluted blood glucose measurements. Anal. Chim. Acta, 183 (I 986) 59-66. [8] i.M. Christie, P.H. Treloar and P. Vadgama, Plasticized poly(vinyl chloride) as a pennselective barrier membrane for high-selectivity amperometric sensors and biosensors. Anal. ChinL Acta, 269 0992) 65-73. [9] !. lliev, P. Atanasov, S. Gamburzev and A. Kaisheva, Transient response of electrochemical biosensors with asymmetrical sandwich membranes, Sensors and Actuators B, 8 (1992) 65-72. li0l T. Schulmeister, Mathematical treatment of concentration profiles and anodic current of amperometric multilayer enzyme electrodes. Anal. Chim. Acta, 198 0987) 223-229. Jill J. Crank, The Mathematics of D~ffusion, Oxford Science, New York, 1990.
Biographies Andrea Cambiaso was born in Geneva in 1958. He received the Laurea degree in Electronic Engineering from the University of Geneva in 1983. Since 1985 he has worked in the Bioelectronics Lab. of DIBE. His main scientific interests are: potentiometric and amperometric biosensors; acoustic microscopy; imaging systems characterization.
Table ! Km al~ values as a function of diffusion coefficient I351 of layer 1. The related Km constant (computed in the hypothesis of a simple bulk reaction) is 4,2 mM. Diffusion coefficient DSI of layer I (cm2 s-!)
Km app (raM)
5E-10 BE-10 l.lE-9
31 25 21
Luca Delfino received the Laurea degree in Electronic Engineering from the University of Geneva in 1994. At present he is a collaborator of the Bioelectronics Lab. of DIBE. His main scientific interests are: potentiometric and amperometric biosensors. Massimo Grattarola is Associate Professor of Bioelectronics at the University of Genoa, where he teaches courses in Bioelectronics and Biomedical Technologies. His research interests include development of cytometric
A. Cambiaso et al. / Sensors and Actuators B 33 (1996) 203-207
techniques (scanning probe microscopies); interfacing networks of neuron with arrays of microelectronics devices; modelling of silicon-based biosensors and their application to monitor the cell microenvironment (i.e., cellular engineering). Giovanni Verreschi received the Laurea degree in Electronic Engineering in 1993 from the University of Genoa. He is working on a Ph.D. degree in Electronics and Computer Science, coordinated by the University of Genova. His research has focused on development of software applications for the analysis and interpretation of bioelectrochemistry signals. He is currently engaged in the Department of Biopysical and Electronic Engineering (DIBE) of the University of Genoa. David Ashworth obtained a Ph.D. in chemistry from the University of Lancaster and spent several years as an industrial chemist. In 1985 he moved to UMIST to carry out research on optical fibre chemical sensors, and in
207
1993 moved to the University of Manchester to coordinate research on fruit monitoring biosensors. Andrew Maines is a postdoctoral Research Associate in the Department of Medicine at the University of Manchester. He obtained a B.Sc. in Marine Biology from the University of Liverpool in 1989 and a Ph.D. in Chemical Engineering from the University of Leeds in 1993. He is currently working on the development of fruit monitoring biosensors. Pankaj Vadgama is Professor of Clinical Biochemistry in the Department of Medicine at the University of Manchester, Honorea-y Consultant Chemical Pathologist at Hope Hospital, and Head of the Department of Medicine. He qualified in Medicine at the University of Newcastleupon-Tyne in 1971, also gaining a degree in Chemistry, and a Ph.D. in 1984. His main interest is in the use of biosensors for clinical monitoring.
ELSEVIER
Sensors and Actuators B 33 (1996) 203-207
CHBMICAL
Modelling and simulation of a diffusion limited glucose biosensor A. Cambiaso a, L. Delfino a, M. Grattarola a,*, G. Verreschia, D. Ashworth b, A. Maines b, P. V a d g a m a b aBioelectronics Laboratory and Bioelectronic Technologies Laboratory, do Advanced Biotechnology Centre, DIBE, Universityof Genoa, via Opera Pia I IA, 16145 Genova, Italy bSection of Clinical Biochemistry, Departmentof Medicine, Hope Hospital, Eccles Old Road, Salford, M6 8HD, UK Accepted 4 March 1996
Abstract A mathematical model for an amperometric enzyme sensor is described as a tool to design glucose biosensors, operating in a diffusion limited regimen. This kind of enzyme electrode really comprises a miniature reactor system rather than a classical form of transducer. The equations constituting the model are obtained taking into account both the diffusion process and the enzymatic reaction in the general (not pseudo-stationary) case. The model is numerically integrated in order to obtain the time behaviour of the electrode current. The Michaelis-Menten apparent constant (Km app) of the whole system is obtained from the simulation and compared with that computed for a bulk system.
Keywords: Amperometric sensors; Diffusion-reactionsystems; Sensor modelling;Glucose; Computer simulations; Biosensor
1. Introduction The development of amperometric enzyme electrodes for the measurement of glucose has been the subject of intense investigation since the construction of the first electrochemical glucose sensor by Clark and Lyons in 1962 [1]. Initially, the major driving force behind this research was the need for clinical glucose monitoring, particularly in relation to the treatment of diabetes mellitus [2]. However, applications have diversified and there is a growing research effort into the monitoring of glucose, and other analytes, in industry [3], agriculture and food [4]. The oxidoreductase enzyme glucose oxidase is commonly used in the development of glucose sensors. The enzyme catalyses the oxidation of glucose: fl-D-glucose + 02 + H - O ~ Doxidase - e l u c o n""i c
acid + H202 (1)
* Corresponding author. Tel.: +39 10 3532761; fax: +39 I0 3532133; e-mail: [email protected]
0925-4005/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved Pli S0925-4005(96)01830-8
In the 'classical' device the consumption of oxygen [5] or the formation of hydrogen peroxide [6] is monitored at an electrode. The concentration of glucose in many matrices greatly exceeds the Km of glucose oxidase (4.2 mmol 1-I) and therefore limits the range over which a linear response can be achieved. This is a particular problem with food applications, where the glucose concentration can be in excess of 500 mmol I-L By the use of external substrate diffusion limiting membranes, the linearity can be greatly enhanced [7]. Amperometric detection of H202 requires a relatively high polarising voltage (ca. +0.65 V versus Ag/AgCl) and as a consequence, sensors based on this detection modality are susceptible to interference from a range of electrooxidisable species found in clinical, industrial and food matrices, including urate and ascorbate. However, interference can be prevented by the interposition of a membrane between the enzyme layer and the electrode which is capable of rejection of interferents whilst allowing diffusion of H202 [8].
204
A. Cambiaso et al./ Sensors and Actuators B 33 (1996) 203-207
2. Sensor model
A generalised glucose enzyme electrode would therefore consist of essentially three layers: the outer diffusion limiting membrane, in contact with the bulk solution containing the analyte of interest; the immobilised enzyme layer, within which substrate is depleted and product is formed; and the inner selective membrane in contact with the electrode [9]. A model of a multiiayer enzyme electrode was already proposed in Ref. [10]. In the present work, the outer membrane performs the dual function of diffusion limitation and selectivity, whilst the inner membrane is a permeable support membrane. Fig. 1 shows the modelled monodimensional structure with the basic interfaces involved in a typical electrode construction. The indicated parameters are referred to the simplified version of the reaction (I), reported in (2).
~
k
___~k+____.~
(2)
,E+P
20 0
layer I
layer3
layer2 E,C
solution
d-
I ¢leclrode
-%
__% d+ a2
X
'
0
'
1000
'
2000 Timebl
3000
Fig. 2. Electrode current transient for different substrate concentrations: (a) 10 mmol i-l, (b) 40 mmol I-!, (c) 70 mmol !-l, (d) 100 mmol! -1.
state assumptions giving the following system of nonlinear equations: aC - - - = KI+EoS-KI+CS-(KI_ + K2+)C at OS 02S - - = D s . . . . . . . KI+EoSI+KI+CS+ Kl_C
Ox 2
OP Ot
bulk
a)
40
--
The considered phenomena are: diffusion of S (substrate = glucose) and P (product) through layer 1, layer 2 and layer 3. Reaction between S and E (enzyme) in layer 2. The behaviour of the system in the different compartments is described in the following, with reference to Fig. 1: X < 0 (bulk solution): the substrate is constant (S - So) and the concentration of product is negligible (P = 0). 0 < X < XI: S and P are free to diffuse. XI
(c)
Bo
0t
E+S~-+C
(d)
100
(3)
O2P = De - -
Ox 2 +
K2+ C
where Ds and De are the diffusion coefficients for S and P in layer 2, respectively. The following boundary conditions were set at each iteration: S = SO, P = 0 in X = 0; O$/Ox =0 (that is flux = 0), P = 0 in X = X3; Ds(OPIOX) and Dp(0P/0X) constant (that is flux constant) through the other interfaces The electrode current is evaluated at the electrode surface by means of Eq. (4): i(t) = nF. A. De -a-P ;:9X (x3,t)
(4)
where A is the electrode area, De is the diffusion coefficient for P in layer 3, and nF is the total molar charge. 3. Computer simulation
The sensor model was numerically integrated by means of a computer program developed in C language on a DEC3000 workstation (Digital, Alpha). The equations related to layers 1 and 3 were discretised using the Crank-Nicholson technique [10]. Such an implicit method gives the advantage of unconditioned numerical stability in the case of simple diffusion equations. For system (3), describing layer 2, an implicit method of integration was developed, starting from the Crank-Nicholson technique. At the starting time S and P were set = 0 in the three layers; E was set equal to Eo in layer 2. 4. Resul~
Fig. l, Qualitative sketch of the one-dimensionai modelled structure.
The simulation program was tested by utilising pa-
205
A. Cambiaso et al. I Sensors and Actuators B 33 (1996) 203-207
x0
x2
xl
x3
100 80 w
60 40
g 20 0
I
I
:
I
',
:
:
:
:
,
|
One dimensional concentration profile (A.U.)
Fig. 3. Substmte concentration profile after (from bottom to top): 900 s, 1800 s, 2700 s, 3600 s.
rameter values evaluated during experiments dealing with the measurements on fruit juice. In pnrticular the diffusion coefficients have been set as follows: layer 1 5.0 x 10-I° cm 2 s-I (S), 1.0 × 10-I! cm 2 s-I (P); layer 2 1.0 x 10-Tcm 2 s-I (S), 5.0 x 10-Tcm 2 s-I (P); layer 3 1.7 x 10-9cm 2 s (S), 8.5 x 10-gem 2 s-! (P) The kinetic constants of the enzymatic reaction in layer 2 have been set: KI+ = 1 2 × 10-2 mmol -I s-I 1, KI_ = 0.68 x 10-3 s-l, /(2+ = 5.0 × 10-2 s-I. The concentration Eo in layer 2 has been set equal to 0.125 mmol 1-l. The thickness of the layers have be set: I 0 / t m for the limiting membrane, 40/~m for the enzymatic layer and 15/tm for the inner membrane. Fig. 2 shows an example of the transient behaviour of the electrode current, computed for 4 different substrate concentrations. There was a very long transient time as expected from this kind of diffusion limiting sensors. Fig. 3 shows the substrate concentration profile through the three layers constituting the model. Four curves are drawn, computed at different simulation times for a substrate concentration S = 100 mmoi i-I. The progressive diffusion of the substrate in the sensor structure can be seen.
3
x0
x2
×1
2.5
!
M
_m
2
=
1.5 =
1 0.5 0
x3
I
I
t
. ,
|
.
,
. ,
,
.
,
. •
,
.
,
.
,
In Fig. 4 the concentration profile of the product is drawn, at different simulation times, under the same conditions as Fig. 3. The slope of these curves at the electrode surface (X3) is proportional to the electrode current. The very long transient time, experimentally obtained with this kind of sensor, suggests that for the evaluation of the substrate concentration the use of techniques alternative to the measurement of the plateau current can be beneficial. For example the maximum of the derivative of the transient curves can be employed for such a purpose (see Fig. 5). The simulation program also gives the opportunity to evaluate the apparent Michaelis-Menten constant Km app taking into account the diffusion effect. The Km for the enzyme in solution is that value of the free uncombined substrate concentration at which half of the enzyme reactive sites are occupied, and the reaction velocity is 50% of the maximum. This assumes that the substrate has free access to the enzyme. If the enzyme and bulk substrate are separated by a diffusion restricting membrane the external substrate concentration can be higher before half of the enzyme sites are occupied, and the apparent Km of the enzyme in this situation is greater than in solution. This apparent Km, denoted as Km app varies as the thickness and diffusion coefficient (Dsi) of the membrane vary, and takes on a different value in each unique situa(d) 0.1
(¢)
0.08 0.06 --= 0.04 0.02
,
,
|
One dimensional concentration profile (A.U.)
0 1000
2000
3000
Time Isl Fig. 4. Product concentration profile after (from bottom to top): 900 s, 1800 s, 2700 s, 3600 s.
Fig. 5. Time derivative of the curves of Fig. 2.
206
A. Cambiaso et aL / Sensors and Actuators B 33 (1996) 203-207 (ci ~
"----'-
Acknowledgements
(a)
Work supported by the European Union, Measurements and testing Programme, BIOSUMO project.
80
References
°
0
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Substrate bulk Ceneentrafion |mMI Fig. 6. Steady state current curves utilised to evaluate the apparent Km, computed for three different diffusion coefficients for S on layer 1: (a) 1.1 x 10-9 cm 2 s"!, (b) g.0 x 10-I0 cm 2 s"l, (c) 5.0 x 10-10 cm2 s -!.
tion. Some simulations have been carried out in order to asses the behaviour of Km app against the variation of the bulk concentration (So). Fig. 6 shows the steady state current as a function of So, computed after 3600 s, for three different values of the diffusion coefficient Dsj of the substrate S in layer I (diffusion limiting membrane). Decreasing the diffusion coefficient can be seen to decrease the response (current) magnitude whilst extending the substrate concentration range over which there is an approximately linear relationship between response and concentration. The Km ~ parameters have been evaluated as the So value corresponding to 50% of the current maximum steady state value. Table 1 shows the calculated values for Km app as a function of layer 1 diffusion coefficient. This demonstrates that decreasing the diffusion coefficient of the outer membrane increases K m app thus extending response linearity,
$. Conclusions The proposed model is a useful tool for the design of diffusion limited amperometric sensors. The simulation program can be used to predict the effect of variation in sensor parameters on sensor response by utilising the information present in the transient period. For example, the effect of membrane thickness and permeability on response sensitivity, velocity and linearity can be predicted. Moreover, new indications can be gained about the values assumed by the kinetic constants when an enzyme reaction is forced to take place near an electrode surface,
[1] L.C. Clark and L. Lyons, Electrode system for continuous monitoring in cardiovascular surgery. Ann. iV. E Acad. ScL, 102 (1962) 29--45. [2l G.D. Velho, G. Reach and D.R. Thevenot, The design and development of in vivo glucosf~ sensors for an artificial endocrine pancreas, in I. Karube and G.S. Wilson (Eds.), Biosensors: Fundaenentals and Applicatio'zs Turner APF, Oxford University Press, New York, 1987, pp. 3cJ0--408. 13] B. Grundig and C. Krabisch, Electron mediator-modified electrode for the deterinination of glucose in fermentation media. Anal, Chim. Act~ 183 (1989) 59--66. [4] J.R. Whitaker, The need for biosensors in the food industry and food research, in G, Wagner and G.G. Guilbault (eds.), Food Bio. sensor Analysis, Marcel Dekkcr, New York, 1994, pp. 13-30. [5] SJ. Updike and G.P. Hicks, The enzyme electrode, Nature, 214 (1967) 986-988. 16] G.G. Guilbanlt, Analytical Uses of Immobilised Enzymes, Marcel Dekker, New York, 1984. [7l W.H. Mullen, F.H. Keedy, S.J. Churchouse and P.M. Vadgama, Glucose enzyme electrode with extended iinearity - application to undiluted blood glucose measurements. Anal. Chim. Acta, 183 (I 986) 59-66. [8] i.M. Christie, P.H. Treloar and P. Vadgama, Plasticized poly(vinyl chloride) as a pennselective barrier membrane for high-selectivity amperometric sensors and biosensors. Anal. ChinL Acta, 269 0992) 65-73. [9] !. lliev, P. Atanasov, S. Gamburzev and A. Kaisheva, Transient response of electrochemical biosensors with asymmetrical sandwich membranes, Sensors and Actuators B, 8 (1992) 65-72. li0l T. Schulmeister, Mathematical treatment of concentration profiles and anodic current of amperometric multilayer enzyme electrodes. Anal. Chim. Acta, 198 0987) 223-229. Jill J. Crank, The Mathematics of D~ffusion, Oxford Science, New York, 1990.
Biographies Andrea Cambiaso was born in Geneva in 1958. He received the Laurea degree in Electronic Engineering from the University of Geneva in 1983. Since 1985 he has worked in the Bioelectronics Lab. of DIBE. His main scientific interests are: potentiometric and amperometric biosensors; acoustic microscopy; imaging systems characterization.
Table ! Km al~ values as a function of diffusion coefficient I351 of layer 1. The related Km constant (computed in the hypothesis of a simple bulk reaction) is 4,2 mM. Diffusion coefficient DSI of layer I (cm2 s-!)
Km app (raM)
5E-10 BE-10 l.lE-9
31 25 21
Luca Delfino received the Laurea degree in Electronic Engineering from the University of Geneva in 1994. At present he is a collaborator of the Bioelectronics Lab. of DIBE. His main scientific interests are: potentiometric and amperometric biosensors. Massimo Grattarola is Associate Professor of Bioelectronics at the University of Genoa, where he teaches courses in Bioelectronics and Biomedical Technologies. His research interests include development of cytometric
A. Cambiaso et al. / Sensors and Actuators B 33 (1996) 203-207
techniques (scanning probe microscopies); interfacing networks of neuron with arrays of microelectronics devices; modelling of silicon-based biosensors and their application to monitor the cell microenvironment (i.e., cellular engineering). Giovanni Verreschi received the Laurea degree in Electronic Engineering in 1993 from the University of Genoa. He is working on a Ph.D. degree in Electronics and Computer Science, coordinated by the University of Genova. His research has focused on development of software applications for the analysis and interpretation of bioelectrochemistry signals. He is currently engaged in the Department of Biopysical and Electronic Engineering (DIBE) of the University of Genoa. David Ashworth obtained a Ph.D. in chemistry from the University of Lancaster and spent several years as an industrial chemist. In 1985 he moved to UMIST to carry out research on optical fibre chemical sensors, and in
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1993 moved to the University of Manchester to coordinate research on fruit monitoring biosensors. Andrew Maines is a postdoctoral Research Associate in the Department of Medicine at the University of Manchester. He obtained a B.Sc. in Marine Biology from the University of Liverpool in 1989 and a Ph.D. in Chemical Engineering from the University of Leeds in 1993. He is currently working on the development of fruit monitoring biosensors. Pankaj Vadgama is Professor of Clinical Biochemistry in the Department of Medicine at the University of Manchester, Honorea-y Consultant Chemical Pathologist at Hope Hospital, and Head of the Department of Medicine. He qualified in Medicine at the University of Newcastleupon-Tyne in 1971, also gaining a degree in Chemistry, and a Ph.D. in 1984. His main interest is in the use of biosensors for clinical monitoring.